Theorem ipasslem1 26465

Description: Lemma for ipassi 26475. Show the inner product associative law for nonnegative integers. (Contributed by NM, 27-Apr-2007.) (New usage is discouraged.)

Hypotheses

Ref	Expression
ip1i.1	|- X = ( BaseSet ` U )
ip1i.2	|- G = ( +v ` U )
ip1i.4	|- S = ( .sOLD ` U )
ip1i.7	|- P = ( .iOLD ` U )
ip1i.9	|- U e. CPreHil OLD
ipasslem1.b	|- B e. X

Assertion

Ref	Expression
ipasslem1	|- ( ( N e. NN0 /\ A e. X ) -> ( ( N S A ) P B ) = ( N x. ( A P B ) ) )

Proof of Theorem ipasslem1

Dummy variables j k are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		nn0cn 10876	. . . . . . . . . . 11 |- ( k e. NN0 -> k e. CC )
2		ax-1cn 9594	. . . . . . . . . . . 12 |- 1 e. CC
3		ip1i.9	. . . . . . . . . . . . . 14 |- U e. CPreHil OLD
4	3	phnvi 26450	. . . . . . . . . . . . 13 |- U e. NrmCVec
5		ip1i.1	. . . . . . . . . . . . . 14 |- X = ( BaseSet ` U )
6		ip1i.2	. . . . . . . . . . . . . 14 |- G = ( +v ` U )
7		ip1i.4	. . . . . . . . . . . . . 14 |- S = ( .sOLD ` U )
8	5, 6, 7	nvdir 26245	. . . . . . . . . . . . 13 |- ( ( U e. NrmCVec /\ ( k e. CC /\ 1 e. CC /\ A e. X ) ) -> ( ( k + 1 ) S A ) = ( ( k S A ) G ( 1 S A ) ) )
9	4, 8	mpan 675	. . . . . . . . . . . 12 |- ( ( k e. CC /\ 1 e. CC /\ A e. X ) -> ( ( k + 1 ) S A ) = ( ( k S A ) G ( 1 S A ) ) )
10	2, 9	mp3an2 1351	. . . . . . . . . . 11 |- ( ( k e. CC /\ A e. X ) -> ( ( k + 1 ) S A ) = ( ( k S A ) G ( 1 S A ) ) )
11	1, 10	sylan 474	. . . . . . . . . 10 |- ( ( k e. NN0 /\ A e. X ) -> ( ( k + 1 ) S A ) = ( ( k S A ) G ( 1 S A ) ) )
12	5, 7	nvsid 26241	. . . . . . . . . . . . 13 |- ( ( U e. NrmCVec /\ A e. X ) -> ( 1 S A ) = A )
13	4, 12	mpan 675	. . . . . . . . . . . 12 |- ( A e. X -> ( 1 S A ) = A )
14	13	adantl 468	. . . . . . . . . . 11 |- ( ( k e. NN0 /\ A e. X ) -> ( 1 S A ) = A )
15	14	oveq2d 6304	. . . . . . . . . 10 |- ( ( k e. NN0 /\ A e. X ) -> ( ( k S A ) G ( 1 S A ) ) = ( ( k S A ) G A ) )
16	11, 15	eqtrd 2484	. . . . . . . . 9 |- ( ( k e. NN0 /\ A e. X ) -> ( ( k + 1 ) S A ) = ( ( k S A ) G A ) )
17	16	oveq1d 6303	. . . . . . . 8 |- ( ( k e. NN0 /\ A e. X ) -> ( ( ( k + 1 ) S A ) P B ) = ( ( ( k S A ) G A ) P B ) )
18		ipasslem1.b	. . . . . . . . . . . . 13 |- B e. X
19		ip1i.7	. . . . . . . . . . . . . 14 |- P = ( .iOLD ` U )
20	5, 19	dipcl 26344	. . . . . . . . . . . . 13 |- ( ( U e. NrmCVec /\ A e. X /\ B e. X ) -> ( A P B ) e. CC )
21	4, 18, 20	mp3an13 1354	. . . . . . . . . . . 12 |- ( A e. X -> ( A P B ) e. CC )
22	21	mulid2d 9658	. . . . . . . . . . 11 |- ( A e. X -> ( 1 x. ( A P B ) ) = ( A P B ) )
23	22	adantl 468	. . . . . . . . . 10 |- ( ( k e. NN0 /\ A e. X ) -> ( 1 x. ( A P B ) ) = ( A P B ) )
24	23	oveq2d 6304	. . . . . . . . 9 |- ( ( k e. NN0 /\ A e. X ) -> ( ( ( k S A ) P B ) + ( 1 x. ( A P B ) ) ) = ( ( ( k S A ) P B ) + ( A P B ) ) )
25	5, 7	nvscl 26240	. . . . . . . . . . . 12 |- ( ( U e. NrmCVec /\ k e. CC /\ A e. X ) -> ( k S A ) e. X )
26	4, 25	mp3an1 1350	. . . . . . . . . . 11 |- ( ( k e. CC /\ A e. X ) -> ( k S A ) e. X )
27	1, 26	sylan 474	. . . . . . . . . 10 |- ( ( k e. NN0 /\ A e. X ) -> ( k S A ) e. X )
28	5, 6, 7, 19, 3	ipdiri 26464	. . . . . . . . . . 11 |- ( ( ( k S A ) e. X /\ A e. X /\ B e. X ) -> ( ( ( k S A ) G A ) P B ) = ( ( ( k S A ) P B ) + ( A P B ) ) )
29	18, 28	mp3an3 1352	. . . . . . . . . 10 |- ( ( ( k S A ) e. X /\ A e. X ) -> ( ( ( k S A ) G A ) P B ) = ( ( ( k S A ) P B ) + ( A P B ) ) )
30	27, 29	sylancom 672	. . . . . . . . 9 |- ( ( k e. NN0 /\ A e. X ) -> ( ( ( k S A ) G A ) P B ) = ( ( ( k S A ) P B ) + ( A P B ) ) )
31	24, 30	eqtr4d 2487	. . . . . . . 8 |- ( ( k e. NN0 /\ A e. X ) -> ( ( ( k S A ) P B ) + ( 1 x. ( A P B ) ) ) = ( ( ( k S A ) G A ) P B ) )
32	17, 31	eqtr4d 2487	. . . . . . 7 |- ( ( k e. NN0 /\ A e. X ) -> ( ( ( k + 1 ) S A ) P B ) = ( ( ( k S A ) P B ) + ( 1 x. ( A P B ) ) ) )
33		oveq1 6295	. . . . . . 7 |- ( ( ( k S A ) P B ) = ( k x. ( A P B ) ) -> ( ( ( k S A ) P B ) + ( 1 x. ( A P B ) ) ) = ( ( k x. ( A P B ) ) + ( 1 x. ( A P B ) ) ) )
34	32, 33	sylan9eq 2504	. . . . . 6 |- ( ( ( k e. NN0 /\ A e. X ) /\ ( ( k S A ) P B ) = ( k x. ( A P B ) ) ) -> ( ( ( k + 1 ) S A ) P B ) = ( ( k x. ( A P B ) ) + ( 1 x. ( A P B ) ) ) )
35		adddir 9631	. . . . . . . . 9 |- ( ( k e. CC /\ 1 e. CC /\ ( A P B ) e. CC ) -> ( ( k + 1 ) x. ( A P B ) ) = ( ( k x. ( A P B ) ) + ( 1 x. ( A P B ) ) ) )
36	2, 35	mp3an2 1351	. . . . . . . 8 |- ( ( k e. CC /\ ( A P B ) e. CC ) -> ( ( k + 1 ) x. ( A P B ) ) = ( ( k x. ( A P B ) ) + ( 1 x. ( A P B ) ) ) )
37	1, 21, 36	syl2an 480	. . . . . . 7 |- ( ( k e. NN0 /\ A e. X ) -> ( ( k + 1 ) x. ( A P B ) ) = ( ( k x. ( A P B ) ) + ( 1 x. ( A P B ) ) ) )
38	37	adantr 467	. . . . . 6 |- ( ( ( k e. NN0 /\ A e. X ) /\ ( ( k S A ) P B ) = ( k x. ( A P B ) ) ) -> ( ( k + 1 ) x. ( A P B ) ) = ( ( k x. ( A P B ) ) + ( 1 x. ( A P B ) ) ) )
39	34, 38	eqtr4d 2487	. . . . 5 |- ( ( ( k e. NN0 /\ A e. X ) /\ ( ( k S A ) P B ) = ( k x. ( A P B ) ) ) -> ( ( ( k + 1 ) S A ) P B ) = ( ( k + 1 ) x. ( A P B ) ) )
40	39	exp31 608	. . . 4 |- ( k e. NN0 -> ( A e. X -> ( ( ( k S A ) P B ) = ( k x. ( A P B ) ) -> ( ( ( k + 1 ) S A ) P B ) = ( ( k + 1 ) x. ( A P B ) ) ) ) )
41	40	a2d 29	. . 3 |- ( k e. NN0 -> ( ( A e. X -> ( ( k S A ) P B ) = ( k x. ( A P B ) ) ) -> ( A e. X -> ( ( ( k + 1 ) S A ) P B ) = ( ( k + 1 ) x. ( A P B ) ) ) ) )
42		eqid 2450	. . . . . 6 |- ( 0vec ` U ) = ( 0vec ` U )
43	5, 42, 19	dip0l 26350	. . . . 5 |- ( ( U e. NrmCVec /\ B e. X ) -> ( ( 0vec ` U ) P B ) = 0 )
44	4, 18, 43	mp2an 677	. . . 4 |- ( ( 0vec ` U ) P B ) = 0
45	5, 7, 42	nv0 26251	. . . . . 6 |- ( ( U e. NrmCVec /\ A e. X ) -> ( 0 S A ) = ( 0vec ` U ) )
46	4, 45	mpan 675	. . . . 5 |- ( A e. X -> ( 0 S A ) = ( 0vec ` U ) )
47	46	oveq1d 6303	. . . 4 |- ( A e. X -> ( ( 0 S A ) P B ) = ( ( 0vec ` U ) P B ) )
48	21	mul02d 9828	. . . 4 |- ( A e. X -> ( 0 x. ( A P B ) ) = 0 )
49	44, 47, 48	3eqtr4a 2510	. . 3 |- ( A e. X -> ( ( 0 S A ) P B ) = ( 0 x. ( A P B ) ) )
50		oveq1 6295	. . . . . 6 |- ( j = 0 -> ( j S A ) = ( 0 S A ) )
51	50	oveq1d 6303	. . . . 5 |- ( j = 0 -> ( ( j S A ) P B ) = ( ( 0 S A ) P B ) )
52		oveq1 6295	. . . . 5 |- ( j = 0 -> ( j x. ( A P B ) ) = ( 0 x. ( A P B ) ) )
53	51, 52	eqeq12d 2465	. . . 4 |- ( j = 0 -> ( ( ( j S A ) P B ) = ( j x. ( A P B ) ) <-> ( ( 0 S A ) P B ) = ( 0 x. ( A P B ) ) ) )
54	53	imbi2d 318	. . 3 |- ( j = 0 -> ( ( A e. X -> ( ( j S A ) P B ) = ( j x. ( A P B ) ) ) <-> ( A e. X -> ( ( 0 S A ) P B ) = ( 0 x. ( A P B ) ) ) ) )
55		oveq1 6295	. . . . . 6 |- ( j = k -> ( j S A ) = ( k S A ) )
56	55	oveq1d 6303	. . . . 5 |- ( j = k -> ( ( j S A ) P B ) = ( ( k S A ) P B ) )
57		oveq1 6295	. . . . 5 |- ( j = k -> ( j x. ( A P B ) ) = ( k x. ( A P B ) ) )
58	56, 57	eqeq12d 2465	. . . 4 |- ( j = k -> ( ( ( j S A ) P B ) = ( j x. ( A P B ) ) <-> ( ( k S A ) P B ) = ( k x. ( A P B ) ) ) )
59	58	imbi2d 318	. . 3 |- ( j = k -> ( ( A e. X -> ( ( j S A ) P B ) = ( j x. ( A P B ) ) ) <-> ( A e. X -> ( ( k S A ) P B ) = ( k x. ( A P B ) ) ) ) )
60		oveq1 6295	. . . . . 6 |- ( j = ( k + 1 ) -> ( j S A ) = ( ( k + 1 ) S A ) )
61	60	oveq1d 6303	. . . . 5 |- ( j = ( k + 1 ) -> ( ( j S A ) P B ) = ( ( ( k + 1 ) S A ) P B ) )
62		oveq1 6295	. . . . 5 |- ( j = ( k + 1 ) -> ( j x. ( A P B ) ) = ( ( k + 1 ) x. ( A P B ) ) )
63	61, 62	eqeq12d 2465	. . . 4 |- ( j = ( k + 1 ) -> ( ( ( j S A ) P B ) = ( j x. ( A P B ) ) <-> ( ( ( k + 1 ) S A ) P B ) = ( ( k + 1 ) x. ( A P B ) ) ) )
64	63	imbi2d 318	. . 3 |- ( j = ( k + 1 ) -> ( ( A e. X -> ( ( j S A ) P B ) = ( j x. ( A P B ) ) ) <-> ( A e. X -> ( ( ( k + 1 ) S A ) P B ) = ( ( k + 1 ) x. ( A P B ) ) ) ) )
65		oveq1 6295	. . . . . 6 |- ( j = N -> ( j S A ) = ( N S A ) )
66	65	oveq1d 6303	. . . . 5 |- ( j = N -> ( ( j S A ) P B ) = ( ( N S A ) P B ) )
67		oveq1 6295	. . . . 5 |- ( j = N -> ( j x. ( A P B ) ) = ( N x. ( A P B ) ) )
68	66, 67	eqeq12d 2465	. . . 4 |- ( j = N -> ( ( ( j S A ) P B ) = ( j x. ( A P B ) ) <-> ( ( N S A ) P B ) = ( N x. ( A P B ) ) ) )
69	68	imbi2d 318	. . 3 |- ( j = N -> ( ( A e. X -> ( ( j S A ) P B ) = ( j x. ( A P B ) ) ) <-> ( A e. X -> ( ( N S A ) P B ) = ( N x. ( A P B ) ) ) ) )
70	41, 49, 54, 59, 64, 69	nn0indALT 11028	. 2 |- ( N e. NN0 -> ( A e. X -> ( ( N S A ) P B ) = ( N x. ( A P B ) ) ) )
71	70	imp 431	1 |- ( ( N e. NN0 /\ A e. X ) -> ( ( N S A ) P B ) = ( N x. ( A P B ) ) )

Syntax hints:   -> wi 4   /\ wa 371   /\ w3a 984   = wceq 1443   e. wcel 1886   ` cfv 5581  (class class class)co 6288   CCcc 9534   0cc0 9536   1c1 9537   + caddc 9539   x. cmul 9541   NN0cn0 10866   NrmCVeccnv 26196   +vcpv 26197   BaseSetcba 26198   .sOLDcns 26199   0veccn0v 26200   .iOLDcdip 26329   CPreHil OLDccphlo 26446

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-8 1888  ax-9 1895  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430  ax-rep 4514  ax-sep 4524  ax-nul 4533  ax-pow 4580  ax-pr 4638  ax-un 6580  ax-inf2 8143  ax-cnex 9592  ax-resscn 9593  ax-1cn 9594  ax-icn 9595  ax-addcl 9596  ax-addrcl 9597  ax-mulcl 9598  ax-mulrcl 9599  ax-mulcom 9600  ax-addass 9601  ax-mulass 9602  ax-distr 9603  ax-i2m1 9604  ax-1ne0 9605  ax-1rid 9606  ax-rnegex 9607  ax-rrecex 9608  ax-cnre 9609  ax-pre-lttri 9610  ax-pre-lttrn 9611  ax-pre-ltadd 9612  ax-pre-mulgt0 9613  ax-pre-sup 9614

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 985  df-3an 986  df-tru 1446  df-fal 1449  df-ex 1663  df-nf 1667  df-sb 1797  df-eu 2302  df-mo 2303  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2580  df-ne 2623  df-nel 2624  df-ral 2741  df-rex 2742  df-reu 2743  df-rmo 2744  df-rab 2745  df-v 3046  df-sbc 3267  df-csb 3363  df-dif 3406  df-un 3408  df-in 3410  df-ss 3417  df-pss 3419  df-nul 3731  df-if 3881  df-pw 3952  df-sn 3968  df-pr 3970  df-tp 3972  df-op 3974  df-uni 4198  df-int 4234  df-iun 4279  df-br 4402  df-opab 4461  df-mpt 4462  df-tr 4497  df-eprel 4744  df-id 4748  df-po 4754  df-so 4755  df-fr 4792  df-se 4793  df-we 4794  df-xp 4839  df-rel 4840  df-cnv 4841  df-co 4842  df-dm 4843  df-rn 4844  df-res 4845  df-ima 4846  df-pred 5379  df-ord 5425  df-on 5426  df-lim 5427  df-suc 5428  df-iota 5545  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-isom 5590  df-riota 6250  df-ov 6291  df-oprab 6292  df-mpt2 6293  df-om 6690  df-1st 6790  df-2nd 6791  df-wrecs 7025  df-recs 7087  df-rdg 7125  df-1o 7179  df-oadd 7183  df-er 7360  df-en 7567  df-dom 7568  df-sdom 7569  df-fin 7570  df-sup 7953  df-oi 8022  df-card 8370  df-pnf 9674  df-mnf 9675  df-xr 9676  df-ltxr 9677  df-le 9678  df-sub 9859  df-neg 9860  df-div 10267  df-nn 10607  df-2 10665  df-3 10666  df-4 10667  df-n0 10867  df-z 10935  df-uz 11157  df-rp 11300  df-fz 11782  df-fzo 11913  df-seq 12211  df-exp 12270  df-hash 12513  df-cj 13155  df-re 13156  df-im 13157  df-sqrt 13291  df-abs 13292  df-clim 13545  df-sum 13746  df-grpo 25912  df-gid 25913  df-ginv 25914  df-ablo 26003  df-vc 26158  df-nv 26204  df-va 26207  df-ba 26208  df-sm 26209  df-0v 26210  df-nmcv 26212  df-dip 26330  df-ph 26447

This theorem is referenced by:  ipasslem2  26466  ipasslem3  26467  ipasslem4  26468